What sorts of calculations called for the invention of logarithm tables?

Question

It's often stated that Napier invented logarithms in order to simplify the task of engineers and scientists who had to crunch numbers by hand. Now, logarithms obviously do simplify calculations (as I learned myself when I turned up at a chemistry exam with no calculator - thankfully chemists give all data in logarithmic form!), in the sense that they reduce multiplication to addition. But in order to do this in practice, you need to:

1. Look up your factors in the logarithm table.
2. Add the factors, which could be quite large or have quite a few decmial places.
3. Look up the result in the logarithm table.

Which I would have thought totally outweighted the time gained by not having to do multiplication, especially considering that you could make a mistake while looking something up, and that even if you don't the result is only going to be an approximation.

Can someone provide a concrete example of a task performed in the era of John Napier which was demonstrably made easier by the use of logarithms? How much easier? Ideally I'd like to see a "before and after", that is an example of the same task (like maybe compiling a trig table) being done with logarithms and without, and how long it took in each case.

Answer 1

From the MacTutor biography of Kepler:

Calculating tables, the normal business for an astronomer, always involved heavy arithmetic. Kepler was accordingly delighted when in 1616 he came across Napier's work on logarithms (published in 1614). However, Mästlin promptly told him first that it was unseemly for a serious mathematician to rejoice over a mere aid to calculation and second that it was unwise to trust logarithms because no-one understood how they worked.... Kepler's answer to the second objection was to publish a proof of how logarithms worked... Kepler calculated tables of eight-figure logarithms, which were published with the Rudolphine Tables (Ulm, 1628).

Note that logarithms also allow the extraction of roots, as Napier noted in his introduction:

Seeing there is nothing (right well-beloved Students of the Mathematics) that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.

If you have do a series of multiplications, divisions, and/or root extractions (e.g., finding a geometric mean), you needn't find the anti-logarithms of the intermediate results, saving more time.

It is not surprising that astronomers were among the first to embrace logarithms. First, astronomy was the most mathematical of all the sciences at that time, and astronomical calculations were quite involved and varied. However, practical problems such as surveying and navigation would also benefit.

Turning to your objections: many of these calculations inherently involve approximations or data of limited accuracy, so you weren't losing anything by using logarithms (provided the tables offered sufficient precision). As for errors, note Napier's complaint about the old-fashioned methods being "subject to many slippery errors". As for time, just try multiplying eight-figure numbers with and without logarithms; you will find it much faster with logs, even taking account of table lookup. (The modern analysis of algorithms tells us that addition, using the standard "schoolbook" method, has time complexity proportional to $n$; for multiplication, the "schoolbook" algorithm has time complexity proportional to $n^2$. Here $n$ is the number of places.)

This webpage goes into detail about one of the problems common to astronomy, navigation, and cartography: solving spherical triangles. Equation like $$\cos a = \cos b \cos c + \sin b \sin c \cos\alpha$$ and $$\sin b \sin\alpha = \sin a \sin \beta$$ make frequent appearances. To quote the webpage, "Preparation of ephemeral tables...demanded reams of tedious calculations like this".

The problem was so acute that Christoph Clavius, a leading astronomer, devised a clever method (called prosthaphaeresis) for reducing multiplication to addition via trig formulas. The idea is to reverse-engineer the addition formulas for sine and cosine to express products of trig values as sums and differences of other trig values. (See the web page for details.) Prosthaphaeresis was used by Tycho Brahe, but was superceded by logarithms, which involved fewer operations.

Final point: it was common in those days to construct tables of compound functions, e.g., $\log(\sin(x))$. (Aside: this fact lead to the empirical discovery of the formula $\int\sec(\phi) d\phi = \log |\sec(\phi) + \tan(\phi)|$, used in Mercator's projection, before its proof via calculus. See Eli Maor's book Trigonometric Delights for the full story.) Thus many uses of logs might have been indirect.